Finding The Exponential Function F(x) = Ca^x Through Points (1,1) And (2,8)

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In this article, we will explore how to determine an exponential function of the form f(x) = ca^x that passes through two given points. This is a fundamental concept in mathematics with applications in various fields, including finance, biology, and physics. Exponential functions are crucial for modeling phenomena involving growth or decay, and understanding how to find these functions from given data points is essential.

We will walk through the process step-by-step, using the specific example of finding the exponential function that contains the points (1, 1) and (2, 8). By the end of this article, you will have a clear understanding of how to solve similar problems and grasp the underlying principles of exponential functions.

Exponential functions are characterized by their rapid growth or decay. The general form of an exponential function is f(x) = ca^x, where:

  • f(x) is the value of the function at x.
  • c is the initial value or the y-intercept (the value of f(x) when x = 0).
  • a is the base, which determines the rate of growth or decay. If a > 1, the function represents exponential growth; if 0 < a < 1, it represents exponential decay.
  • x is the independent variable, often representing time.

To uniquely define an exponential function, we need to determine the values of both c and a. Having two points on the graph of the function provides us with two equations, which we can solve simultaneously to find these values. This process is akin to solving a system of linear equations, but with the added complexity of dealing with exponential terms.

Given two points (x_1, y_1) and (x_2, y_2) on the graph of the exponential function f(x) = ca^x, we can set up two equations by substituting the coordinates of the points into the function:

  1. y_1 = ca^{x_1}
  2. y_2 = ca^{x_2}

In our case, the given points are (1, 1) and (2, 8). Substituting these values, we get:

  1. 1 = ca^1 or 1 = ca
  2. 8 = ca^2

Now we have a system of two equations with two unknowns, c and a. The next step is to solve this system to find the values of c and a.

To solve the system of equations, we can use a method called substitution or division. In this case, division is a more straightforward approach. We divide the second equation by the first equation:

(8 = ca^2) / (1 = ca)

This simplifies to:

8/1 = (ca^2) / (ca)

8 = a

Now that we have the value of a, which is 8, we can substitute it back into one of the original equations to find c. Let's use the first equation, 1 = ca:

1 = c(8)

Divide both sides by 8:

c = 1/8

So, we have found that a = 8 and c = 1/8. Now we can write the exponential function.

Having found the values of c and a, we can now write the exponential function that passes through the points (1, 1) and (2, 8). Recall the general form of the exponential function:

f(x) = ca^x

Substitute the values we found:

f(x) = (1/8) * 8^x

This is the exponential function that contains the two given points. To verify this, we can plug in the x-values of the given points and check if the function yields the corresponding y-values.

For x = 1:

f(1) = (1/8) * 8^1 = (1/8) * 8 = 1

For x = 2:

f(2) = (1/8) * 8^2 = (1/8) * 64 = 8

Both points satisfy the function, so our solution is correct. The exponential function that passes through (1, 1) and (2, 8) is f(x) = (1/8) * 8^x.

Exponential functions are incredibly versatile and have numerous applications across various disciplines. Understanding how to derive these functions from given data points allows us to model and predict real-world phenomena accurately.

  1. Financial Growth: Exponential functions are fundamental in finance for modeling compound interest. For instance, if you invest an initial amount of money at a fixed interest rate compounded annually, the growth of your investment over time can be modeled using an exponential function. The base a would be (1 + r), where r is the interest rate, and c would be the initial investment.

  2. Population Growth: In biology, exponential functions are used to model population growth under ideal conditions (unlimited resources and no predators). The base a would represent the growth rate, and c would be the initial population size. However, it’s important to note that in real-world scenarios, population growth is often limited by factors such as resource availability, leading to more complex models like logistic growth.

  3. Radioactive Decay: Exponential decay is another significant application. Radioactive materials decay at a rate proportional to the amount present. The decay is modeled by an exponential function where a is less than 1, representing the decay factor, and c is the initial amount of the radioactive substance. This is crucial in fields like nuclear physics and archaeology (e.g., carbon dating).

  4. Spread of Diseases: Exponential functions can also model the initial spread of infectious diseases. In the early stages of an outbreak, the number of infected individuals can grow exponentially, assuming each infected person infects a certain number of others. However, this model is simplified and real-world epidemics often have more complex dynamics.

  5. Learning Curves: In psychology and education, exponential functions are sometimes used to model learning curves. For example, the rate at which a person learns a new skill might initially be rapid but then slow down over time as they approach their maximum proficiency.

  6. Cooling Processes: Newton's Law of Cooling states that the rate at which an object cools is proportional to the temperature difference between the object and its surroundings. This can be modeled using an exponential function, where the temperature difference decreases exponentially over time.

  7. Marketing and Sales: Exponential functions can be used to model the growth of sales or market penetration of a new product. Initially, sales might grow rapidly as early adopters purchase the product, but this growth may slow down as the market becomes saturated.

These examples illustrate the broad applicability of exponential functions in modeling various phenomena. Understanding how to derive these functions from data, as we’ve done in this article, is a powerful tool for analysis and prediction.

In conclusion, finding an exponential function of the form f(x) = ca^x that passes through two given points involves setting up and solving a system of equations. By substituting the coordinates of the points into the general form of the exponential function, we create two equations with two unknowns, c and a. Solving this system, often by division and substitution, allows us to determine the specific values of c and a and construct the exponential function. In our example, the function that contains the points (1, 1) and (2, 8) is f(x) = (1/8) * 8^x.

This skill is crucial for modeling various real-world phenomena, from financial growth and population dynamics to radioactive decay and the spread of diseases. Mastering the process of finding exponential functions from data points enhances our ability to understand and predict the behavior of complex systems.

By understanding and applying these principles, you can confidently tackle problems involving exponential functions and appreciate their significance in mathematics and its applications. The ability to model real-world phenomena using mathematical functions is a cornerstone of scientific and engineering disciplines.

The function which contains the two graph points (1, 1) and (2, 8) is f(x) = (1/8) * 8^x.